From a point A on level ground, the angle of elevation to a water tower is 17°. From point B, 60 m nearer the tower, the angle of elevation to the top is 21°. Find the height of the tower to the nearest metre. [90m]

What I've done:

<C = 180 - (17+21)
<C = 142°

SinC/c = SinA/a
a = Sin17(60/Sin142)
a = 28.4934

Sin21 = h/a
Sin21 = h/28.4934
h = 28.4934(Sin21)
=10.2

What am I doing wrong?

I don't understand you work, except the first angle. Now you determine the other angle in that outer triangle. From that you should use law of cosines to solve for the slant height from the inner point to the top of tower.

60/sintop angle=innerslantheight/sin17
solve for innerslant height.

Now you have height of tower in
h/slantheight=sin21
and solve for height h.

what you are doing wrong is ignoring the cotangent function.

If you check your diagram, you will see that the height h is found using

h cot17° - h cot21° = 60
h = 90.1 m

Based on the information you have provided, it seems like you have made a calculation error when solving for the height of the tower using the trigonometric ratios.

To find the height of the tower correctly, you can follow these steps:

1. Draw a diagram to visualize the situation. Label points A, B, and the top of the tower as T. Also label the distances AB and AT.
2. Use the information given to write down the relationships between the angles and distances. From point A, the angle of elevation to the tower is 17°. From point B, the angle of elevation to the top of the tower is 21°. Point B is 60 meters closer to the tower, which means AB is 60 meters.
3. Notice that you have a right triangle formed by points A, T (top of the tower), and the foot of the tower (let's call it C). You can apply trigonometry to this triangle.
4. Let's solve for the height of the tower (h). It is the side opposite the angle of 17° in triangle ATC.
5. Write down the trigonometric equation for the triangle ATC. We have Sin(17°) = h / AT.
6. Now, we need to find AT. To do that, notice that AB = AT - BT. Since AB is given as 60 meters and BT is the distance from point B to the top of the tower, we need to find BT.
7. In triangle BCT, BT is the adjacent side to the angle of 21°. We can use the tangent ratio: Tan(21°) = BT / AB.
8. Rearrange this equation to solve for BT: BT = AB * Tan(21°).
9. Substitute the value from step 8 into the one from step 6: AT = AB + BT = AB + AB * Tan(21°).
10. Now, substitute the value from step 9 into the equation from step 5: Sin(17°) = h / (AB + AB * Tan(21°)).
11. Solve this equation for h by multiplying both sides by (AB + AB * Tan(21°)): h = Sin(17°) * (AB + AB * Tan(21°)).

By plugging in the values you have, Sin(17°) ≈ 0.29237 and AB = 60, you can calculate:

h ≈ 0.29237 * (60 + 60 * Tan(21°)) ≈ 90 meters.

Therefore, the height of the tower is approximately 90 meters, which matches the answer you provided.